Complex Numbers
nth Roots
MMTS_Full_Test_04
Grade 12
Question:
If $1,z_1,z_2,\ldots,z_{n-1}$ are $n$-th roots of unity, then $(1-z_1)(1-z_2)\cdots(1-z_{n-1})$ equals
Step-by-Step Solution
Key Concept: $x^n-1=(x-1)(x-z_1)\cdots(x-z_{n-1})$; put $x=1$
$\prod_{k=1}^{n-1}(1-z_k)=\lim_{x\to 1}\frac{x^n-1}{x-1}=n$.
Correct Answer: 2